A353358 Expansion of e.g.f. exp(log(1 - x)^4).

1, 0, 0, 0, 24, 240, 2040, 17640, 182616, 2340576, 34907520, 567732000, 9811675104, 179804319552, 3507724531584, 72964001073600, 1614757714491456, 37860036000293376, 936291898320463872, 24333527620574701056, 662723505438520771584, 18871765275000834201600

Offset: 0

Author

Seiichi Manyama, May 06 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..438

Crossrefs

Column k=4 of A357882.

Cf. A000142, A353344, A353404.

Cf. A347003, A353119, A353665.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1-x)^4)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(log(1-x)^3)))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i-1, j-1)*abs(stirling(j, 4, 1))*v[i-j+1])); v;

PARI

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/k!);

Formula

E.g.f.: (1 - x)^((log(1 - x))^3).

a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).

a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/k!.